1.INTRODUCTION present some background material in the appendices,that novice and expert alike can consult;most material in the appendices is brought without proof,although the details of some specialized computations are provided. Keeping in mind our stated emphasis on accessibility over generality,the book is essentially divided in two parts.In Chapters 2 and 3,we present a self contained analysis of random matrices,quickly focusing on the Gaussian ensembles and culminating in the derivation of the gap probabilities at 0 and the Tracy-Widom law.These chapters can be read with very little background knowledge,and are particularly suitable for an introductory study.In the second part of the book, Chapters 4 and 5,we use more advanced techniques,requiring more extensive background,to emphasize and generalize certain aspects of the theory,and to introduce the theory of free probability. So what is a random matrix,and what questions are we about to study?Through- out,let F=R or F=C,and set B=1 in the former case and B=2 in the latter.(In Section 4.1,we will also consider the case F=H,the skew-field of quaternions, see Appendix E for definitions and details.)Let Matw(F)denote the space of N -by-N matrices with entries in,and let denote the subset of self-adjoint matrices (i.e.,real symmetric if B=I and Hermitian if B=2.)One can always consider the sets Maty(F)andB-1,2,as submanifolds of an appropriate Euclidean space,and equip it with the induced topology and(Borel)sigma-field. Recall that a probability space is a triple (,P)so that is a sigma-algebra of subsets of and P is a probability measure on(,).In that setting,a random matrix Xy is a measurable map from (,to Matw(F). Our main object of interest are the eigemalues of random matrices.Recall that the eigenvalues of a matrix HE Maty(F)are the roots of the characteristic polynomial P(=)=det(=IN-H),with IN the identity matrix.Therefore,on the (open)set where the eigenvalues are all simple,they are smooth functions of the entries of Yy (a more complete discussion can be found in Section 4.1) We will be mostly concered in this book with self-adjoint matrices HB) B=1,2,in which case the eigenvalues are all real and can be ordered.Thus, for HE光)，we let(l≤…≤2w(l)be the eigenvalues of H.Acon- sequence of the perturbation theory of normal matrices (see Lemma A.4)is that the eigenvalues {(H)}are continuous functions in H(this also follows from the Hoffman-Wielandt theorem,Theorem 2.1.19).In particular,if Xy is a random matrix then the eigenvalues (X)}are random variables. We present now a guided tour of the book.We begin by considering in Chap- ter 2 Wigner matrices.Those are symmetric (or Hermitian)matrices Xy whose
2 1. INTRODUCTION present some background material in the appendices, that novice and expert alike can consult; most material in the appendices is brought without proof, although the details of some specialized computations are provided. Keeping in mind our stated emphasis on accessibility over generality, the book is essentially divided in two parts. In Chapters 2 and 3, we present a self contained analysis of random matrices, quickly focusing on the Gaussian ensembles and culminating in the derivation of the gap probabilities at 0 and the Tracy–Widom law. These chapters can be read with very little background knowledge, and are particularly suitable for an introductory study. In the second part of the book, Chapters 4 and 5, we use more advanced techniques, requiring more extensive background, to emphasize and generalize certain aspects of the theory, and to introduce the theory of free probability. So what is a random matrix, and what questions are we about to study? Throughout, let F = R or F = C, and set β = 1 in the former case and β = 2 in the latter. (In Section 4.1, we will also consider the case F = H, the skew-field of quaternions, see Appendix E for definitions and details.) Let MatN(F) denote the space of N -by -N matrices with entries in F, and let H (β) N denote the subset of self-adjoint matrices (i.e., real symmetric if β = 1 and Hermitian if β = 2.) One can always consider the sets MatN(F) and H (β) N , β = 1,2, as submanifolds of an appropriate Euclidean space, and equip it with the induced topology and (Borel) sigma-field. Recall that a probability space is a triple (Ω,F,P) so that F is a sigma-algebra of subsets of Ω and P is a probability measure on (Ω,F). In that setting, a random matrix XN is a measurable map from (Ω,F) to MatN(F). Our main object of interest are the eigenvalues of random matrices. Recall that the eigenvalues of a matrix H ∈ MatN(F) are the roots of the characteristic polynomial PN(z) = det(zIN − H), with IN the identity matrix. Therefore, on the (open) set where the eigenvalues are all simple, they are smooth functions of the entries of XN (a more complete discussion can be found in Section 4.1). We will be mostly concerned in this book with self-adjoint matrices H ∈ H (β) N , β = 1,2, in which case the eigenvalues are all real and can be ordered. Thus, for H ∈ H (β) N , we let λ1(H) ≤ ··· ≤ λN(H) be the eigenvalues of H. A consequence of the perturbation theory of normal matrices (see Lemma A.4) is that the eigenvalues {λi(H)} are continuous functions in H (this also follows from the Hoffman–Wielandt theorem, Theorem 2.1.19). In particular, if XN is a random matrix then the eigenvalues {λi(XN)} are random variables. We present now a guided tour of the book. We begin by considering in Chapter 2 Wigner matrices. Those are symmetric (or Hermitian) matrices XN whose
1.INTRODUCTION entries are independent and identically distributed,except for the symmetry con- straints.For xER,let o,denote the Dirac measure at x,i.e the unique proba- bility measure satisfying ffd&r=f(x)for all continuous functions on R.Let LN-N-)denote the empirical measure of the eigenvalues of Xy. Wigner's Theorem(Theorem 2.1.1)asserts that,under appropriate assumptions on the law of the entries,Lw converges(with respect to the weak convergence of measures)towards a deterministic probability measure,the semicircle law.We present in Chapter 2 several proofs of Wigner's Theorem.The first,in Section 2.1,involves a combinatorial machinery,that is also exploited to yield central limit theorems and estimates on the spectral radius of XN.After first introducing in Section 2.3 some useful estimates on the deviation between the empirical mea- sure and its mean,we define in Section 2.4 the Stieltjes transform of measures and use it to give another quick proof of Wigner's theorem. Having discussed techniques valid for entries distributed according to general laws,we turn attention to special situations involving additional symmetry.The simplest of these concerns the Gaussian ensembles,the GOE and GUE.so named because their law is invariant under conjugation by orthogonal(resp.,unitary) matrices.The latter extra symmetry is crucial in deriving in Section 2.5 an explicit joint distribution for the eigenvalues(thus,effectively reducing consideration from a problem involving order of N2 random variables,namely the matrix entries,to ones involving only N variables).(The GSE,or Gaussian symplectic ensemble, also shares this property and is discussed briefly.)A large deviations principle for the empirical distribution,which leads to yet another proof of Wigner's Theorem, follows in Section 2.6. The expression for the joint density of the eigenvalues in the Gaussian ensem- bles is the starting point for obtaining local information on the eigenvalues.This is the topic of Chapter 3.The bulk of the chapter deals with the GUE,because in that situation the eigenvalues form a determinantal process.This allows one to effectively represent the probability that no eigenvalues are present in a set as a Fredholm determinant,a notion that is particularly amenable to asymptotic analysis.Thus,after representing in Section 3.2 the joint density for the GUE in terms of a determinant involving appropriate orthogonal polynomials,the Hermite polynomials,we develop in Section 3.4 in an elementary way some aspects of the theory of Fredholm determinants.We then present in Section 3.5 the asymptotic analysis required in order to study the gap probability at 0,that is the probabil- ity that no eigenvalue is present in an interval around the origin.Relevant tools, such as the Laplace method,are developed along the way.Section 3.7 repeats this analysis for the edge of the spectrum,introducing along the way the method of
1. INTRODUCTION 3 entries are independent and identically distributed, except for the symmetry constraints. For x ∈ R, let δx denote the Dirac measure at x, i.e the unique probability measure satisfying R f dδx = f(x) for all continuous functions on R. Let LN = N −1 ∑ N i=1 δλi(XN) denote the empirical measure of the eigenvalues of XN. Wigner’s Theorem (Theorem 2.1.1) asserts that, under appropriate assumptions on the law of the entries, LN converges (with respect to the weak convergence of measures) towards a deterministic probability measure, the semicircle law. We present in Chapter 2 several proofs of Wigner’s Theorem. The first, in Section 2.1, involves a combinatorial machinery, that is also exploited to yield central limit theorems and estimates on the spectral radius of XN. After first introducing in Section 2.3 some useful estimates on the deviation between the empirical measure and its mean, we define in Section 2.4 the Stieltjes transform of measures and use it to give another quick proof of Wigner’s theorem. Having discussed techniques valid for entries distributed according to general laws, we turn attention to special situations involving additional symmetry. The simplest of these concerns the Gaussian ensembles, the GOE and GUE, so named because their law is invariant under conjugation by orthogonal (resp., unitary) matrices. The latter extra symmetry is crucial in deriving in Section 2.5 an explicit joint distribution for the eigenvalues (thus, effectively reducing consideration from a problem involving order of N 2 random variables, namely the matrix entries, to ones involving only N variables). (The GSE, or Gaussian symplectic ensemble, also shares this property and is discussed briefly.) A large deviations principle for the empirical distribution, which leads to yet another proof of Wigner’s Theorem, follows in Section 2.6. The expression for the joint density of the eigenvalues in the Gaussian ensembles is the starting point for obtaining local information on the eigenvalues. This is the topic of Chapter 3. The bulk of the chapter deals with the GUE, because in that situation the eigenvalues form a determinantal process. This allows one to effectively represent the probability that no eigenvalues are present in a set as a Fredholm determinant, a notion that is particularly amenable to asymptotic analysis. Thus, after representing in Section 3.2 the joint density for the GUE in terms of a determinant involving appropriate orthogonal polynomials, the Hermite polynomials, we develop in Section 3.4 in an elementary way some aspects of the theory of Fredholm determinants. We then present in Section 3.5 the asymptotic analysis required in order to study the gap probability at 0, that is the probability that no eigenvalue is present in an interval around the origin. Relevant tools, such as the Laplace method, are developed along the way. Section 3.7 repeats this analysis for the edge of the spectrum, introducing along the way the method of
1.INTRODUCTION steepest descent.The link with integrable systems and the Painleve eguations is established in Sections 3.6 and 3.8. As mentioned before,the eigenvalues of the GUE are an example of a deter- minantal process.The other Gaussian ensembles (GOE and GSE)do not fall into this class,but they do enjoy a structure where certain pfaffians replace determi- nants.This leads to a considerable more involved analysis,the details of which are provided in Section 3.9. Chapter 4 is a hodge-podge collection of general tools and results,whose com- mon feature is that they all require some new tools.We begin in Section 4.1 with a re-derivation of the joint law of the eigenvalues of the Gaussian ensemble,in a geometric framework based on Lie theory.We use this framework to derive the expressions for the joint distribution of eigenvalues of Wishart matrices,as well as random matrices from the various unitary groups,and random projectors. Section 4.2 studies in some depth determinantal processes,including their con- struction,associated central limit theorems,convergence and ergodic properties. Section 4.3 studies what happens when in the GUE(or GOE),the Gaussian entries are replaced by Brownian motions.The powerful tools of stochastic analysis can then be brought to bear and lead to functional laws of large numbers,central limit theorems,and large deviations.Section 4.4 consists of an in-depth treatment of concentration techniques and their application to random matrices;it is a general- ization of the discussion in the short Section 2.3.Finally,in Section 4.5,we study a family of tri-diagonal matrices,parametrized by a parameter B,whose distribu- tion of eigenvalues coincides with that of members of the Gaussian ensembles for B=1,2,4.The study of the maximal eigenvalue for this family is linked to the spectrum of an appropriate random Schroedinger operator. Chapter 5 is devoted to free probability theory,a probability theory for certain noncommutative variables,equipped with a notion of independence called free independence.Invented in the early 1990s.free probability theory has become a versatile tool in analyzing the law of non-commutative polynomials in random matrices,and of the limits of the empirical measure of eigenvalues of polynomials in several random matrices.We develop the necessary preliminaries and defini- tions in Section 5.2,introduce free independence in Section 5.3.and discuss the link with random matrices in Section 5.4.We conclude the chapter with Section 5.5,that studies the convergence of the spectral radius of non-commutative poly- nomials of random matrices. Each chapter ends with bibliography notes.These are not meant to be com- prehensive,but rather guide the reader through the enormous literature and give some hint of recent developments.Although we have tried to represent accurately
4 1. INTRODUCTION steepest descent. The link with integrable systems and the Painleve equations ´ is established in Sections 3.6 and 3.8. As mentioned before, the eigenvalues of the GUE are an example of a determinantal process. The other Gaussian ensembles (GOE and GSE) do not fall into this class, but they do enjoy a structure where certain pfaffians replace determinants. This leads to a considerable more involved analysis, the details of which are provided in Section 3.9. Chapter 4 is a hodge-podge collection of general tools and results, whose common feature is that they all require some new tools. We begin in Section 4.1 with a re-derivation of the joint law of the eigenvalues of the Gaussian ensemble, in a geometric framework based on Lie theory. We use this framework to derive the expressions for the joint distribution of eigenvalues of Wishart matrices, as well as random matrices from the various unitary groups, and random projectors. Section 4.2 studies in some depth determinantal processes, including their construction, associated central limit theorems, convergence and ergodic properties. Section 4.3 studies what happens when in the GUE (or GOE), the Gaussian entries are replaced by Brownian motions. The powerful tools of stochastic analysis can then be brought to bear and lead to functional laws of large numbers, central limit theorems, and large deviations. Section 4.4 consists of an in-depth treatment of concentration techniques and their application to random matrices; it is a generalization of the discussion in the short Section 2.3. Finally, in Section 4.5, we study a family of tri-diagonal matrices, parametrized by a parameter β, whose distribution of eigenvalues coincides with that of members of the Gaussian ensembles for β = 1,2,4. The study of the maximal eigenvalue for this family is linked to the spectrum of an appropriate random Schroedinger operator. Chapter 5 is devoted to free probability theory, a probability theory for certain noncommutative variables, equipped with a notion of independence called free independence. Invented in the early 1990s, free probability theory has become a versatile tool in analyzing the law of non-commutative polynomials in random matrices, and of the limits of the empirical measure of eigenvalues of polynomials in several random matrices. We develop the necessary preliminaries and definitions in Section 5.2, introduce free independence in Section 5.3, and discuss the link with random matrices in Section 5.4. We conclude the chapter with Section 5.5, that studies the convergence of the spectral radius of non-commutative polynomials of random matrices. Each chapter ends with bibliography notes. These are not meant to be comprehensive, but rather guide the reader through the enormous literature and give some hint of recent developments. Although we have tried to represent accurately
1.INTRODUCTION 5 the historical development of the subject,we have necessarily omitted important references,misrepresented facts,or plainly erred.Our apologies to those authors whose work we have thus unintentionally slighted Of course,we have barely scratched the surface of human knowledge concern- ing random matrices.We mention now the most glaring omissions,together with references to some recent books that cover these topics.We have not discussed the theory of the Riemann-Hilbert problem and its relation to integrable systems, Painleve equations,asymptotics of orthogonal polynomials and random matrices. The interested reader is referred to the books [FolKN06],[Dei99]and [DeG09] for an in-depth treatment.We do not discuss the relation between asymptotics of random matrices and combinatorial problems-a good summary of these ap- pears in [BaDS08].We barely discuss applications of random matrices,and in particular do not review the recent increase in applications to statistics or com- munication theory-for a nice introduction to the latter we refer to [TuV04].We have presented only a partial discussion of ensembles of matrices that possess ex- plicit joint distribution of eigenvalues.For a more complete discussion,including also the case of non-Hermitian matrices that are not unitary,we refer the reader to [For05].Finally,we have not touched at the link between random matrices and number theory;the interested reader should consult [KaS99]for a taste of that link.We further refer to the bibliography notes for additional reading,less glaring omissions,and references
1. INTRODUCTION 5 the historical development of the subject, we have necessarily omitted important references, misrepresented facts, or plainly erred. Our apologies to those authors whose work we have thus unintentionally slighted. Of course, we have barely scratched the surface of human knowledge concerning random matrices. We mention now the most glaring omissions, together with references to some recent books that cover these topics. We have not discussed the theory of the Riemann–Hilbert problem and its relation to integrable systems, Painlev´e equations, asymptotics of orthogonal polynomials and random matrices. The interested reader is referred to the books [FoIKN06], [Dei99] and [DeG09] for an in-depth treatment. We do not discuss the relation between asymptotics of random matrices and combinatorial problems – a good summary of these appears in [BaDS08]. We barely discuss applications of random matrices, and in particular do not review the recent increase in applications to statistics or communication theory – for a nice introduction to the latter we refer to [TuV04]. We have presented only a partial discussion of ensembles of matrices that possess explicit joint distribution of eigenvalues. For a more complete discussion, including also the case of non-Hermitian matrices that are not unitary, we refer the reader to [For05]. Finally, we have not touched at the link between random matrices and number theory; the interested reader should consult [KaS99] for a taste of that link. We further refer to the bibliography notes for additional reading, less glaring omissions, and references
2 Real and Complex Wigner matrices 2.1 Real Wigner matrices:traces,moments and combinatorics We introduce in this section a basic model of random matrices.Nowhere do we attempt to provide the weakest assumptions or sharpest results available.We point out in the bibliographical notes(Section 2.7)some places where the interested reader can find finer results. Start with two independent families of i.i.d.,zero mean,real-valued random variables [Zi)isi<and [Y)is,such that EZ12 =1 and,for all integersk>1, k=max(EZ2,El<∞ (2.1.1) Consider the(symmetric)N x N matrix XN with entries wU,0=x60={Z,ii<, Yi/VN,ifi=j. (2.1.2) We call such a matrix a Wigner matrix,and if the random variables Zi.and Yi are Gaussian,we use the term Gaussian Wigner matrix.The case of Gaussian Wigner matrices in which EY2=2 is of particular importance,and for reasons that will become clearer in Chapter 3,such matrices(rescaled by vN)are referred to as GOE(Gaussian Orthogonal Ensemble)matrices. Let元denote the(real)eigenvalues of X,.with≤≤.≤，and define the empirical distribution of the eigenvalues as the (random)probability measure on R defined by =8 Define the standard semicircle distribution as the probability distribution o(x)dx 6
2 Real and Complex Wigner matrices 2.1 Real Wigner matrices: traces, moments and combinatorics We introduce in this section a basic model of random matrices. Nowhere do we attempt to provide the weakest assumptions or sharpest results available. We point out in the bibliographical notes (Section 2.7) some places where the interested reader can find finer results. Start with two independent families of i.i.d., zero mean, real-valued random variables {Zi, j}1≤i<j and {Yi}1≤i , such that EZ2 1,2 = 1 and, for all integers k ≥ 1, rk := max E|Z1,2| k ,E|Y1| k < ∞. (2.1.1) Consider the (symmetric) N ×N matrix XN with entries XN(j,i) = XN(i, j) = Zi, j/ √ N, if i < j, Yi/ √ N, if i = j. (2.1.2) We call such a matrix a Wigner matrix, and if the random variables Zi, j and Yi are Gaussian, we use the term Gaussian Wigner matrix. The case of Gaussian Wigner matrices in which EY2 1 = 2 is of particular importance, and for reasons that will become clearer in Chapter 3, such matrices (rescaled by √ N) are referred to as GOE (Gaussian Orthogonal Ensemble) matrices. Let λ N i denote the (real) eigenvalues of XN, with λ N 1 ≤ λ N 2 ≤ ... ≤ λ N N , and define the empirical distribution of the eigenvalues as the (random) probability measure on R defined by LN = 1 N N ∑ i=1 δλ N i . Define the standard semicircle distribution as the probability distribution σ(x)dx 6